## A new class of linear operators on $$\ell^2$$ and Schur multipliers for them.(English)Zbl 1146.47017

Let $$B(\ell^2)$$ be the set of all bounded linear operators on $$\ell^2$$ and $$B_w(\ell^2)=\{A=(a_{ij})_{i,j\geq 1}: Ax\in\ell^2$$ for all $$x\in\ell^2$$ with $$| x_k| \downarrow 0\}$$. Notice that $$B_w(\ell^2)$$ contains unbounded operators. For two matrices $$A=(a_{ij})_{i,j\geq 1}$$ and $$B=(b_{ij})_{i,j\geq 1}$$, their Schur product is defined by $$A*B=(a_{ij}\cdot b_{ij})_{i,j\geq 1}$$. Let $$X$$ be either $$B(\ell^2)$$ or $$B_w(\ell^2)$$. The space of Schur multipliers on $$X$$ is the set of all matrices $$M$$ such that $$M*B\in X$$ for all $$B\in X$$. The main result of the paper says that $$B(\ell^2)\subset M(B_w(\ell^2))$$. Let $$T$$ be the set of all infinite Toeplitz matrices. It is also proved that $$M(B_w(\ell^2))\cap T\subset M(B(\ell^2))\cap T$$.

### MSC:

 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators

### Keywords:

Schur multiplier; Toeplitz matrix
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